Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. 2x - 5———— < 5 4

Verified step by step guidance

1. Step 1: Start by isolating the variable term. Multiply both sides of the inequality by 4 to eliminate the fraction: \( 2x - 5 < 20 \).
2. Step 2: Add 5 to both sides of the inequality to further isolate the term with the variable: \( 2x < 25 \).
3. Step 3: Divide both sides by 2 to solve for \( x \): \( x < \frac{25}{2} \).
4. Step 4: Express the solution in interval notation. Since \( x \) is less than \( \frac{25}{2} \), the solution set is \((-\infty, \frac{25}{2})\).
5. Step 5: Verify the solution by testing a value from the solution set in the original inequality to ensure it holds true.

Key Concepts

Inequalities

Inequalities express a relationship where one quantity is larger or smaller than another. In this case, the inequality involves a rational expression, which requires understanding how to manipulate both sides of the inequality to isolate the variable. Solving inequalities often involves similar steps to solving equations, but special attention must be paid to the direction of the inequality sign, especially when multiplying or dividing by negative numbers.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed interval) or excluded (open interval). For example, the interval (2, 5] includes all numbers greater than 2 and up to 5, including 5 but not 2. Understanding how to express solution sets in interval notation is crucial for clearly communicating the results of solving inequalities.

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. In the given inequality, the expression 2x - 5/4 must be analyzed to determine its behavior and the values of x that satisfy the inequality. Understanding how to simplify and manipulate rational expressions is essential for solving inequalities involving them, as it often requires finding common denominators and identifying restrictions on the variable.